论文标题
计算几乎没有网状顶点的系统发育网络：精确的枚举和校正
Counting Phylogenetic Networks with Few Reticulation Vertices: Exact Enumeration and Corrections
论文作者
论文摘要
在以前的工作中，我们给出了$ k $网状网络数量和普通网络的渐近计数结果，并以$ k = 1,2,3 $的计数序列的$ k $网络和明确的指数生成函数。本说明的目的是两个方面。首先，我们对以前的方法进行了一些更正，这些方法过度估算了上述数字，从而给出了错误的指数生成函数（但是，过度估算不会影响我们的渐近计数结果）。其次，我们使用（校正的）指数生成功能来得出$ k = 1,2,3 $网状网络的树木和普通网络数量的显式公式。这种重新介绍了Carona和Zhang的最新结果，以$ k = 2 $为正常网络回答了他们的问题，并在情况下添加了新的配方$ k = 3 $。
In previous work, we gave asymptotic counting results for the number of tree-child and normal networks with $k$ reticulation vertices and explicit exponential generating functions of the counting sequences for $k=1,2,3$. The purpose of this note is two-fold. First, we make some corrections to our previous approach which overcounted the above numbers and thus gives erroneous exponential generating functions (however, the overcounting does not effect our asymptotic counting results). Secondly, we use our (corrected) exponential generating functions to derive explicit formulas for the number of tree-child and normal networks with $k=1,2,3$ reticulation vertices. This re-derives recent results of Carona and Zhang, answers their question for normal networks with $k=2$, and adds new formulas in the case $k=3$.
